Problem Statement of Series 3, Year 40

A cubic container of an edge length $a=1.0\,\mathrm{m}$ is filled with a water bath at $t = 0.0\,\mathrm{^\circ\mskip-2mu\mathup{C}}$. The water surface sits $h=5.0\,\mathrm{mm}$ below the container's upper edge. An ice cube containing an internal cavity of $V_{\mathrm{cavity}} = 0.50\,\mathrm{l}$ floats in the bath. The system receives exactly enough energy to melt the ice completely, and the water remains at its starting temperature. How much water overflows the edge of the container? The air surrounding the bath and filling the cavity is at normal atmospheric pressure and at the same temperature $t$.

Imagine that we lived on a hypothetical planet with half the radius of Earth but the same mass and the same total angular momentum. What would be the magnitude of the gravitational acceleration experienced by bodies on the equator of this planet?

One of the mechanisms used by high-currents circuit breakers is the disconnection of conductors through Ampère's law. The current flowing through the circuit breaker generates a magnetic field that can disconnect the circuit when the current is sufficiently high. Let us consider a simplified model of this mechanism in the situation shown in the figure. The current flowing through the conductors, indicated by the arrows, causes the two vertical conductors to repel one another. The first conductor, on the left in the figure, is fixed, whereas the second can rotate about an axis perpendicular to the plane of the conductors.

Suppose that we increase the current sufficiently slowly so that, at every position, the movable part is in torque equilibrium. The contacts remain electrically connected until the part rotates through an angle $\alpha$, at which point they separate; this angle satisfies $l\sin\alpha \ll d$. How stiff must the spring be, that is, what must its spring constant $k$ be, if we know that the circuit disconnects at the threshold current $I$?

Consider a truncated-cone-shaped mountain with height $H$ and lower radius $R$, whose slopes make an angle $\alpha$ with the ground. A bus is to transport tourists up the mountain; when fully loaded, it has mass $m$, and its wheels have a coefficient of static friction $k$ with respect to the ground. What is the shortest route the bus can take? Assume that the road may follow any trajectory on the mountain's surface and that its inclination is equal to the inclination of the tangent to this trajectory relative to the ground.

Bonus: Find the specific form (parametric representation) of the ideal trajectory in Cartesian coordinates.

Astronauts on the ISS conducted experiments with a bubble in weightlessness. They inflated a drop of water with density $\rho$ and surface tension $\sigma$ into a bubble with radius $R_0$ and wall thickness $h\ll R_0$. Over time, dust particles landed on the bubble, giving it a uniformly distributed charge $Q$. In addition, a small gust of wind imparted momentum to it, causing it to begin expanding and contracting. Although the astronauts could no longer continue the experiment, they were able to observe the bubble's beautiful small oscillations. Calculate their period. Assume that no heat is exchanged between the water and the air inside the bubble. Also assume that the surrounding air is at standard pressure $p_{\mathrm{a}} = 1\,\mathrm{atm}$. Consider the change in the bubble's equilibrium radius due to the charge to be negligible.

This problem has an open solution, so be sure to cite all sources used.

In a greenhouse, we can maintain a higher temperature inside than in the surrounding environment. What would happen if we built another greenhouse inside it? Would this result in an even higher temperature? What if we added yet another one? Determine the theoretical upper limit of the temperature that can be achieved in this way. Also estimate to what extent further increases in temperature remain economically worthwhile.

Investigate the shapes and dimensions of water droplets after they land on an inclined plane as a function of the height from which the droplet is released and the inclination of the plane. Release the water one drop at a time from a dropper to ensure that the droplets have the same volume.